3.2.34 \(\int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx\) [134]

Optimal. Leaf size=212 \[ -\frac {i (c+d x)^2}{3 a^2 f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {4 i d^2 \text {PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]

[Out]

-1/3*I*(d*x+c)^2/a^2/f+4/3*d*(d*x+c)*ln(1+exp(I*(f*x+e)))/a^2/f^2-4/3*I*d^2*polylog(2,-exp(I*(f*x+e)))/a^2/f^3
-1/3*d*(d*x+c)*sec(1/2*f*x+1/2*e)^2/a^2/f^2+2/3*d^2*tan(1/2*f*x+1/2*e)/a^2/f^3+1/3*(d*x+c)^2*tan(1/2*f*x+1/2*e
)/a^2/f+1/6*(d*x+c)^2*sec(1/2*f*x+1/2*e)^2*tan(1/2*f*x+1/2*e)/a^2/f

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Rubi [A]
time = 0.17, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271, 3852, 8, 4269, 3800, 2221, 2317, 2438} \begin {gather*} \frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {i (c+d x)^2}{3 a^2 f}-\frac {4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + a*Cos[e + f*x])^2,x]

[Out]

((-1/3*I)*(c + d*x)^2)/(a^2*f) + (4*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(3*a^2*f^2) - (((4*I)/3)*d^2*PolyLog
[2, -E^(I*(e + f*x))])/(a^2*f^3) - (d*(c + d*x)*Sec[e/2 + (f*x)/2]^2)/(3*a^2*f^2) + (2*d^2*Tan[e/2 + (f*x)/2])
/(3*a^2*f^3) + ((c + d*x)^2*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/
2])/(6*a^2*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rubi steps

\begin {align*} \int \frac {(c+d x)^2}{(a+a \cos (e+f x))^2} \, dx &=\frac {\int (c+d x)^2 \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}\\ &=-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int 1 \, dx,x,-\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^3}-\frac {(2 d) \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(4 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (4 d^2\right ) \int \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=-\frac {i (c+d x)^2}{3 a^2 f}+\frac {4 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {4 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 212, normalized size = 1.00 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (-2 d f (c+d x) \cos \left (\frac {1}{2} (e+f x)\right )-2 i f (c+d x) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (f (c+d x)+4 i d \log \left (1+e^{i (e+f x)}\right )\right )-8 i d^2 \cos ^3\left (\frac {1}{2} (e+f x)\right ) \text {PolyLog}\left (2,-e^{i (e+f x)}\right )+\left (2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )+\left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cos (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 a^2 f^3 (1+\cos (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + a*Cos[e + f*x])^2,x]

[Out]

(2*Cos[(e + f*x)/2]*(-2*d*f*(c + d*x)*Cos[(e + f*x)/2] - (2*I)*f*(c + d*x)*Cos[(e + f*x)/2]^3*(f*(c + d*x) + (
4*I)*d*Log[1 + E^(I*(e + f*x))]) - (8*I)*d^2*Cos[(e + f*x)/2]^3*PolyLog[2, -E^(I*(e + f*x))] + (2*(c^2*f^2 + 2
*c*d*f^2*x + d^2*(1 + f^2*x^2)) + (c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Cos[e + f*x])*Sin[(e + f*x)/2]))
/(3*a^2*f^3*(1 + Cos[e + f*x])^2)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (172 ) = 344\).
time = 0.40, size = 358, normalized size = 1.69

method	result	size
risch	\(\frac {2 i \left (2 i d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}+3 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+2 i c d f \,{\mathrm e}^{2 i \left (f x +e \right )}+2 i d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}+6 c d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+d^{2} x^{2} f^{2}+2 i c d f \,{\mathrm e}^{i \left (f x +e \right )}+3 c^{2} f^{2} {\mathrm e}^{i \left (f x +e \right )}+2 c d \,f^{2} x +c^{2} f^{2}+2 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+4 d^{2} {\mathrm e}^{i \left (f x +e \right )}+2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{2}}-\frac {2 i d^{2} x^{2}}{3 a^{2} f}-\frac {4 i d^{2} e x}{3 a^{2} f^{2}}-\frac {2 i d^{2} e^{2}}{3 a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{3 a^{2} f^{2}}-\frac {4 i d^{2} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}\)	\(358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a+a*cos(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/3*I*(2*I*d^2*f*x*exp(2*I*(f*x+e))+3*d^2*f^2*x^2*exp(I*(f*x+e))+2*I*c*d*f*exp(2*I*(f*x+e))+2*I*d^2*f*x*exp(I*
(f*x+e))+6*c*d*f^2*x*exp(I*(f*x+e))+d^2*x^2*f^2+2*I*c*d*f*exp(I*(f*x+e))+3*c^2*f^2*exp(I*(f*x+e))+2*c*d*f^2*x+
c^2*f^2+2*d^2*exp(2*I*(f*x+e))+4*d^2*exp(I*(f*x+e))+2*d^2)/f^3/a^2/(exp(I*(f*x+e))+1)^3+4/3/a^2*d/f^2*c*ln(exp
(I*(f*x+e))+1)-4/3/a^2*d/f^2*c*ln(exp(I*(f*x+e)))-2/3*I/a^2*d^2/f*x^2-4/3*I/a^2*d^2/f^2*e*x-2/3*I/a^2*d^2/f^3*
e^2+4/3/a^2*d^2/f^2*ln(exp(I*(f*x+e))+1)*x-4/3*I*d^2*polylog(2,-exp(I*(f*x+e)))/a^2/f^3+4/3/a^2*d^2/f^3*e*ln(e
xp(I*(f*x+e)))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 812 vs. \(2 (176) = 352\).
time = 0.59, size = 812, normalized size = 3.83 \begin {gather*} \frac {2 \, {\left (c^{2} f^{2} + 2 \, d^{2} + 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) - {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) - {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x\right )} \cos \left (3 \, f x + 3 \, e\right ) - {\left (3 \, d^{2} f^{2} x^{2} - 2 i \, c d f - 2 \, d^{2} + 2 \, {\left (3 \, c d f^{2} - i \, d^{2} f\right )} x\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 \, c^{2} f^{2} + 2 i \, d^{2} f x + 2 i \, c d f + 4 \, d^{2}\right )} \cos \left (f x + e\right ) - 2 \, {\left (d^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, d^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, d^{2} \cos \left (f x + e\right ) + i \, d^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 i \, d^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 i \, d^{2} \sin \left (f x + e\right ) + d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) - {\left (i \, d^{2} f x + i \, c d f + {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (f x + e\right ) - {\left (d^{2} f x + c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) - {\left (i \, d^{2} f^{2} x^{2} + 2 i \, c d f^{2} x\right )} \sin \left (3 \, f x + 3 \, e\right ) - {\left (3 i \, d^{2} f^{2} x^{2} + 2 \, c d f - 2 i \, d^{2} + 2 \, {\left (3 i \, c d f^{2} + d^{2} f\right )} x\right )} \sin \left (2 \, f x + 2 \, e\right ) - {\left (-3 i \, c^{2} f^{2} + 2 \, d^{2} f x + 2 \, c d f - 4 i \, d^{2}\right )} \sin \left (f x + e\right )\right )}}{-3 i \, a^{2} f^{3} \cos \left (3 \, f x + 3 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (2 \, f x + 2 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (f x + e\right ) + 3 \, a^{2} f^{3} \sin \left (3 \, f x + 3 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (f x + e\right ) - 3 i \, a^{2} f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*cos(f*x+e))^2,x, algorithm="maxima")

[Out]

2*(c^2*f^2 + 2*d^2 + 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(3*f*x + 3*e) + 3*(d^2*f*x + c*d*f)*cos(2*f*x +
2*e) + 3*(d^2*f*x + c*d*f)*cos(f*x + e) - (-I*d^2*f*x - I*c*d*f)*sin(3*f*x + 3*e) - 3*(-I*d^2*f*x - I*c*d*f)*
sin(2*f*x + 2*e) - 3*(-I*d^2*f*x - I*c*d*f)*sin(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (d^2*f^2*x
^2 + 2*c*d*f^2*x)*cos(3*f*x + 3*e) - (3*d^2*f^2*x^2 - 2*I*c*d*f - 2*d^2 + 2*(3*c*d*f^2 - I*d^2*f)*x)*cos(2*f*x
+ 2*e) + (3*c^2*f^2 + 2*I*d^2*f*x + 2*I*c*d*f + 4*d^2)*cos(f*x + e) - 2*(d^2*cos(3*f*x + 3*e) + 3*d^2*cos(2*f
*x + 2*e) + 3*d^2*cos(f*x + e) + I*d^2*sin(3*f*x + 3*e) + 3*I*d^2*sin(2*f*x + 2*e) + 3*I*d^2*sin(f*x + e) + d^
2)*dilog(-e^(I*f*x + I*e)) - (I*d^2*f*x + I*c*d*f + (I*d^2*f*x + I*c*d*f)*cos(3*f*x + 3*e) + 3*(I*d^2*f*x + I*
c*d*f)*cos(2*f*x + 2*e) + 3*(I*d^2*f*x + I*c*d*f)*cos(f*x + e) - (d^2*f*x + c*d*f)*sin(3*f*x + 3*e) - 3*(d^2*f
*x + c*d*f)*sin(2*f*x + 2*e) - 3*(d^2*f*x + c*d*f)*sin(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f
*x + e) + 1) - (I*d^2*f^2*x^2 + 2*I*c*d*f^2*x)*sin(3*f*x + 3*e) - (3*I*d^2*f^2*x^2 + 2*c*d*f - 2*I*d^2 + 2*(3*
I*c*d*f^2 + d^2*f)*x)*sin(2*f*x + 2*e) - (-3*I*c^2*f^2 + 2*d^2*f*x + 2*c*d*f - 4*I*d^2)*sin(f*x + e))/(-3*I*a^
2*f^3*cos(3*f*x + 3*e) - 9*I*a^2*f^3*cos(2*f*x + 2*e) - 9*I*a^2*f^3*cos(f*x + e) + 3*a^2*f^3*sin(3*f*x + 3*e)
+ 9*a^2*f^3*sin(2*f*x + 2*e) + 9*a^2*f^3*sin(f*x + e) - 3*I*a^2*f^3)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (176) = 352\).
time = 0.41, size = 411, normalized size = 1.94 \begin {gather*} -\frac {2 \, d^{2} f x + 2 \, c d f + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) + 2 \, {\left (-i \, d^{2} \cos \left (f x + e\right )^{2} - 2 i \, d^{2} \cos \left (f x + e\right ) - i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (i \, d^{2} \cos \left (f x + e\right )^{2} + 2 i \, d^{2} \cos \left (f x + e\right ) + i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - {\left (2 \, d^{2} f^{2} x^{2} + 4 \, c d f^{2} x + 2 \, c^{2} f^{2} + 2 \, d^{2} + {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cos \left (f x + e\right ) + a^{2} f^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*cos(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/3*(2*d^2*f*x + 2*c*d*f + 2*(d^2*f*x + c*d*f)*cos(f*x + e) + 2*(-I*d^2*cos(f*x + e)^2 - 2*I*d^2*cos(f*x + e)
- I*d^2)*dilog(-cos(f*x + e) + I*sin(f*x + e)) + 2*(I*d^2*cos(f*x + e)^2 + 2*I*d^2*cos(f*x + e) + I*d^2)*dilo
g(-cos(f*x + e) - I*sin(f*x + e)) - 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(f*x + e)^2 + 2*(d^2*f*x + c*d*f
)*cos(f*x + e))*log(cos(f*x + e) + I*sin(f*x + e) + 1) - 2*(d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(f*x + e)^2
+ 2*(d^2*f*x + c*d*f)*cos(f*x + e))*log(cos(f*x + e) - I*sin(f*x + e) + 1) - (2*d^2*f^2*x^2 + 4*c*d*f^2*x + 2
*c^2*f^2 + 2*d^2 + (d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2)*cos(f*x + e))*sin(f*x + e))/(a^2*f^3*cos(f*x
+ e)^2 + 2*a^2*f^3*cos(f*x + e) + a^2*f^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {c^{2}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\cos ^{2}{\left (e + f x \right )} + 2 \cos {\left (e + f x \right )} + 1}\, dx}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a+a*cos(f*x+e))**2,x)

[Out]

(Integral(c**2/(cos(e + f*x)**2 + 2*cos(e + f*x) + 1), x) + Integral(d**2*x**2/(cos(e + f*x)**2 + 2*cos(e + f*
x) + 1), x) + Integral(2*c*d*x/(cos(e + f*x)**2 + 2*cos(e + f*x) + 1), x))/a**2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a+a*cos(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^2/(a*cos(f*x + e) + a)^2, x)

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(a + a*cos(e + f*x))^2,x)

[Out]

\text{Hanged}

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